I flew many air tests at Valley in the early 80s. There was no call in the Air Test Schedule to do a supersonic run. After a timed climb to FL400, it was all gently downhill until the high speed run al low-level - nominally at 2000ft but sometimes a tadge lower!
There was no supersonic run on the FJ course (which I never did [the course]). But I was introduced to them by a venerable colleague who swore that the Hawk did not make a sonic bang; so my first experiences were directed at fields of sheep on the Lleyn.
I was not convinced so whenever I had a go, I checked that I was out to sea and got radar clearance for a "high speed" run. It got to be known around the students and I had no shortage of volunteers to fill the back seat.
It was pretty straight forward: half roll, pull to 45+ degrees nose down and half roll again. Handling and recovery were unremarkable.
The low-level run was more problematic. On some aeroplanes, 167 168 and 169 particularly, approaching 500 kts the turn and slip ball would edge out, and quite a bit or rudder was needed to correct. I took one to Dunsfold to be looked at and remember Andy Jones, the Hawk development TP, approaching the tailcone with a large screwdriver. Swapping the tailcone for the modded one with the large fillet and vertical rear largely eliminated the problem.